#include "blaswrap.h" /* -- translated by f2c (version 19990503). You must link the resulting object file with the libraries: -lf2c -lm (in that order) */ #include "f2c.h" /* Table of constant values */ static complex c_b1 = {0.f,0.f}; static integer c__1 = 1; /* Subroutine */ int cstegr_(char *jobz, char *range, integer *n, real *d__, real *e, real *vl, real *vu, integer *il, integer *iu, real *abstol, integer *m, real *w, complex *z__, integer *ldz, integer *isuppz, real *work, integer *lwork, integer *iwork, integer *liwork, integer * info) { /* System generated locals */ integer z_dim1, z_offset, i__1, i__2; real r__1, r__2; /* Builtin functions */ double sqrt(doublereal); /* Local variables */ static integer iend; static real rmin, rmax; static integer itmp; static real tnrm; static integer i__, j; static real scale; extern logical lsame_(char *, char *); static integer iinfo; extern /* Subroutine */ int sscal_(integer *, real *, real *, integer *), cswap_(integer *, complex *, integer *, complex *, integer *); static integer lwmin; static logical wantz; static integer jj; static logical alleig, indeig; static integer ibegin, iindbl; static logical valeig; extern doublereal slamch_(char *); extern /* Subroutine */ int claset_(char *, integer *, integer *, complex *, complex *, complex *, integer *); static real safmin; extern /* Subroutine */ int xerbla_(char *, integer *); static real bignum; static integer iindwk, indgrs, indwof; extern /* Subroutine */ int clarrv_(integer *, real *, real *, integer *, integer *, real *, integer *, real *, real *, complex *, integer * , integer *, real *, integer *, integer *), slarre_(integer *, real *, real *, real *, integer *, integer *, integer *, real *, real *, real *, real *, integer *); static real thresh; static integer iinspl, indwrk, liwmin; extern doublereal slanst_(char *, integer *, real *, real *); static integer nsplit; static real smlnum; static logical lquery; static real eps, tol, tmp; #define z___subscr(a_1,a_2) (a_2)*z_dim1 + a_1 #define z___ref(a_1,a_2) z__[z___subscr(a_1,a_2)] /* -- LAPACK computational routine (version 3.0) -- Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., Courant Institute, Argonne National Lab, and Rice University October 31, 1999 Purpose ======= CSTEGR computes selected eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix T. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues. The eigenvalues are computed by the dqds algorithm, while orthogonal eigenvectors are computed from various ``good'' L D L^T representations (also known as Relatively Robust Representations). Gram-Schmidt orthogonalization is avoided as far as possible. More specifically, the various steps of the algorithm are as follows. For the i-th unreduced block of T, (a) Compute T - sigma_i = L_i D_i L_i^T, such that L_i D_i L_i^T is a relatively robust representation, (b) Compute the eigenvalues, lambda_j, of L_i D_i L_i^T to high relative accuracy by the dqds algorithm, (c) If there is a cluster of close eigenvalues, "choose" sigma_i close to the cluster, and go to step (a), (d) Given the approximate eigenvalue lambda_j of L_i D_i L_i^T, compute the corresponding eigenvector by forming a rank-revealing twisted factorization. The desired accuracy of the output can be specified by the input parameter ABSTOL. For more details, see "A new O(n^2) algorithm for the symmetric tridiagonal eigenvalue/eigenvector problem", by Inderjit Dhillon, Computer Science Division Technical Report No. UCB/CSD-97-971, UC Berkeley, May 1997. Note 1 : Currently CSTEGR is only set up to find ALL the n eigenvalues and eigenvectors of T in O(n^2) time Note 2 : Currently the routine CSTEIN is called when an appropriate sigma_i cannot be chosen in step (c) above. CSTEIN invokes modified Gram-Schmidt when eigenvalues are close. Note 3 : CSTEGR works only on machines which follow ieee-754 floating-point standard in their handling of infinities and NaNs. Normal execution of CSTEGR may create NaNs and infinities and hence may abort due to a floating point exception in environments which do not conform to the ieee standard. Arguments ========= JOBZ (input) CHARACTER*1 = 'N': Compute eigenvalues only; = 'V': Compute eigenvalues and eigenvectors. RANGE (input) CHARACTER*1 = 'A': all eigenvalues will be found. = 'V': all eigenvalues in the half-open interval (VL,VU] will be found. = 'I': the IL-th through IU-th eigenvalues will be found. ********* Only RANGE = 'A' is currently supported ********************* N (input) INTEGER The order of the matrix. N >= 0. D (input/output) REAL array, dimension (N) On entry, the n diagonal elements of the tridiagonal matrix T. On exit, D is overwritten. E (input/output) REAL array, dimension (N) On entry, the (n-1) subdiagonal elements of the tridiagonal matrix T in elements 1 to N-1 of E; E(N) need not be set. On exit, E is overwritten. VL (input) REAL VU (input) REAL If RANGE='V', the lower and upper bounds of the interval to be searched for eigenvalues. VL < VU. Not referenced if RANGE = 'A' or 'I'. IL (input) INTEGER IU (input) INTEGER If RANGE='I', the indices (in ascending order) of the smallest and largest eigenvalues to be returned. 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not referenced if RANGE = 'A' or 'V'. ABSTOL (input) REAL The absolute error tolerance for the eigenvalues/eigenvectors. IF JOBZ = 'V', the eigenvalues and eigenvectors output have residual norms bounded by ABSTOL, and the dot products between different eigenvectors are bounded by ABSTOL. If ABSTOL is less than N*EPS*|T|, then N*EPS*|T| will be used in its place, where EPS is the machine precision and |T| is the 1-norm of the tridiagonal matrix. The eigenvalues are computed to an accuracy of EPS*|T| irrespective of ABSTOL. If high relative accuracy is important, set ABSTOL to DLAMCH( 'Safe minimum' ). See Barlow and Demmel "Computing Accurate Eigensystems of Scaled Diagonally Dominant Matrices", LAPACK Working Note #7 for a discussion of which matrices define their eigenvalues to high relative accuracy. M (output) INTEGER The total number of eigenvalues found. 0 <= M <= N. If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1. W (output) REAL array, dimension (N) The first M elements contain the selected eigenvalues in ascending order. Z (output) COMPLEX array, dimension (LDZ, max(1,M) ) If JOBZ = 'V', then if INFO = 0, the first M columns of Z contain the orthonormal eigenvectors of the matrix T corresponding to the selected eigenvalues, with the i-th column of Z holding the eigenvector associated with W(i). If JOBZ = 'N', then Z is not referenced. Note: the user must ensure that at least max(1,M) columns are supplied in the array Z; if RANGE = 'V', the exact value of M is not known in advance and an upper bound must be used. LDZ (input) INTEGER The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N). ISUPPZ (output) INTEGER ARRAY, dimension ( 2*max(1,M) ) The support of the eigenvectors in Z, i.e., the indices indicating the nonzero elements in Z. The i-th eigenvector is nonzero only in elements ISUPPZ( 2*i-1 ) through ISUPPZ( 2*i ). WORK (workspace/output) REAL array, dimension (LWORK) On exit, if INFO = 0, WORK(1) returns the optimal (and minimal) LWORK. LWORK (input) INTEGER The dimension of the array WORK. LWORK >= max(1,18*N) If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA. IWORK (workspace/output) INTEGER array, dimension (LIWORK) On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK. LIWORK (input) INTEGER The dimension of the array IWORK. LIWORK >= max(1,10*N) If LIWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the IWORK array, returns this value as the first entry of the IWORK array, and no error message related to LIWORK is issued by XERBLA. INFO (output) INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: if INFO = 1, internal error in SLARRE, if INFO = 2, internal error in CLARRV. Further Details =============== Based on contributions by Inderjit Dhillon, IBM Almaden, USA Osni Marques, LBNL/NERSC, USA Ken Stanley, Computer Science Division, University of California at Berkeley, USA ===================================================================== Test the input parameters. Parameter adjustments */ --d__; --e; --w; z_dim1 = *ldz; z_offset = 1 + z_dim1 * 1; z__ -= z_offset; --isuppz; --work; --iwork; /* Function Body */ wantz = lsame_(jobz, "V"); alleig = lsame_(range, "A"); valeig = lsame_(range, "V"); indeig = lsame_(range, "I"); lquery = *lwork == -1 || *liwork == -1; lwmin = *n * 18; liwmin = *n * 10; *info = 0; if (! (wantz || lsame_(jobz, "N"))) { *info = -1; } else if (! (alleig || valeig || indeig)) { *info = -2; /* The following two lines need to be removed once the RANGE = 'V' and RANGE = 'I' options are provided. */ } else if (valeig || indeig) { *info = -2; } else if (*n < 0) { *info = -3; } else if (valeig && *n > 0 && *vu <= *vl) { *info = -7; } else if (indeig && *il < 1) { *info = -8; /* The following change should be made in DSTEVX also, otherwise IL can be specified as N+1 and IU as N. ELSE IF( INDEIG .AND. ( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) ) THEN */ } else if (indeig && (*iu < *il || *iu > *n)) { *info = -9; } else if (*ldz < 1 || wantz && *ldz < *n) { *info = -14; } else if (*lwork < lwmin && ! lquery) { *info = -17; } else if (*liwork < liwmin && ! lquery) { *info = -19; } if (*info == 0) { work[1] = (real) lwmin; iwork[1] = liwmin; } if (*info != 0) { i__1 = -(*info); xerbla_("CSTEGR", &i__1); return 0; } else if (lquery) { return 0; } /* Quick return if possible */ *m = 0; if (*n == 0) { return 0; } if (*n == 1) { if (alleig || indeig) { *m = 1; w[1] = d__[1]; } else { if (*vl < d__[1] && *vu >= d__[1]) { *m = 1; w[1] = d__[1]; } } if (wantz) { i__1 = z___subscr(1, 1); z__[i__1].r = 1.f, z__[i__1].i = 0.f; } return 0; } /* Get machine constants. */ safmin = slamch_("Safe minimum"); eps = slamch_("Precision"); smlnum = safmin / eps; bignum = 1.f / smlnum; rmin = sqrt(smlnum); /* Computing MIN */ r__1 = sqrt(bignum), r__2 = 1.f / sqrt(sqrt(safmin)); rmax = dmin(r__1,r__2); /* Scale matrix to allowable range, if necessary. */ scale = 1.f; tnrm = slanst_("M", n, &d__[1], &e[1]); if (tnrm > 0.f && tnrm < rmin) { scale = rmin / tnrm; } else if (tnrm > rmax) { scale = rmax / tnrm; } if (scale != 1.f) { sscal_(n, &scale, &d__[1], &c__1); i__1 = *n - 1; sscal_(&i__1, &scale, &e[1], &c__1); tnrm *= scale; } indgrs = 1; indwof = (*n << 1) + 1; indwrk = *n * 3 + 1; iinspl = 1; iindbl = *n + 1; iindwk = (*n << 1) + 1; claset_("Full", n, n, &c_b1, &c_b1, &z__[z_offset], ldz); /* Compute the desired eigenvalues of the tridiagonal after splitting into smaller subblocks if the corresponding of-diagonal elements are small */ thresh = eps * tnrm; slarre_(n, &d__[1], &e[1], &thresh, &nsplit, &iwork[iinspl], m, &w[1], & work[indwof], &work[indgrs], &work[indwrk], &iinfo); if (iinfo != 0) { *info = 1; return 0; } if (wantz) { /* Compute the desired eigenvectors corresponding to the computed eigenvalues Computing MAX */ r__1 = *abstol, r__2 = (real) (*n) * thresh; tol = dmax(r__1,r__2); ibegin = 1; i__1 = nsplit; for (i__ = 1; i__ <= i__1; ++i__) { iend = iwork[iinspl + i__ - 1]; i__2 = iend; for (j = ibegin; j <= i__2; ++j) { iwork[iindbl + j - 1] = i__; /* L10: */ } ibegin = iend + 1; /* L20: */ } clarrv_(n, &d__[1], &e[1], &iwork[iinspl], m, &w[1], &iwork[iindbl], & work[indgrs], &tol, &z__[z_offset], ldz, &isuppz[1], &work[ indwrk], &iwork[iindwk], &iinfo); if (iinfo != 0) { *info = 2; return 0; } } ibegin = 1; i__1 = nsplit; for (i__ = 1; i__ <= i__1; ++i__) { iend = iwork[iinspl + i__ - 1]; i__2 = iend; for (j = ibegin; j <= i__2; ++j) { w[j] += work[indwof + i__ - 1]; /* L30: */ } ibegin = iend + 1; /* L40: */ } /* If matrix was scaled, then rescale eigenvalues appropriately. */ if (scale != 1.f) { r__1 = 1.f / scale; sscal_(m, &r__1, &w[1], &c__1); } /* If eigenvalues are not in order, then sort them, along with eigenvectors. */ if (nsplit > 1) { i__1 = *m - 1; for (j = 1; j <= i__1; ++j) { i__ = 0; tmp = w[j]; i__2 = *m; for (jj = j + 1; jj <= i__2; ++jj) { if (w[jj] < tmp) { i__ = jj; tmp = w[jj]; } /* L50: */ } if (i__ != 0) { w[i__] = w[j]; w[j] = tmp; if (wantz) { cswap_(n, &z___ref(1, i__), &c__1, &z___ref(1, j), &c__1); itmp = isuppz[(i__ << 1) - 1]; isuppz[(i__ << 1) - 1] = isuppz[(j << 1) - 1]; isuppz[(j << 1) - 1] = itmp; itmp = isuppz[i__ * 2]; isuppz[i__ * 2] = isuppz[j * 2]; isuppz[j * 2] = itmp; } } /* L60: */ } } work[1] = (real) lwmin; iwork[1] = liwmin; return 0; /* End of CSTEGR */ } /* cstegr_ */ #undef z___ref #undef z___subscr .